Classical inequalities

Abstract

We use the notation introduced in the previous sections. In particular, let ℝⁿ be again euclidean n-space where n ∈ ℕ. The Schwartz space S (ℝⁿ), its dual S′ (ℝⁿ) and the spaces L _p (ℝⁿ) with 0 < p ≤ ∞ have the same meaning as in 2.1, the latter quasi-normed by (2.1). Let L ^loc₁ (ℝⁿ) be the collection of all complex-valued locally Lebesgue-integrable functions in ℝⁿ. Any f ∈ L ^loc₁ (ℝⁿ) is interpreted in the usual way as a regular distribution. Conversely, as usual, a distribution on ℝⁿ is called regular if, and only if, it can be identified (as a distribution) with a locally integrable function on ℝⁿ. If A(ℝⁿ) is a collection of distributions on ℝⁿ, then

$$A(\mathbb{R}^n ) \subset L_1^{loc} (\mathbb{R}^n ) $$

((11.))

simply means that any element f of A(ℝⁿ) is a regular distribution f ∈ L ^loc₁ (ℝⁿ). Then, in particular, the distribution function μf(λ), the rearrangement f ^*(t) and its maximal function f ^**(t) in (10.2)–(10.4) make sense accepting that they might be infinite. If A ₁(ℝⁿ) and A ₂(ℝⁿ) are two quasi-normed spaces, continuously embedded in S′(ℝⁿ), then

$$A_1 (\mathbb{R}^n ) \subset A_2 (\mathbb{R}^n ) $$

((11.2))

always means that there is a constant c > 0 such that

$$\left\| {f|A_2 (\mathbb{R}^n )} \right\| \leqslant c\left\| {f|A_1 (\mathbb{R}^n )} \right\| for all f \in A_1 (\mathbb{R}^n ), $$

((11.3))

(continuous embedding). On the other hand we do not use the word embedding in connection with inequalities of type (10.12).